Betting trip game

ABSTRACT

A method to play a game of chance using a map of sites as playing surface with moving pieces called movers. The game rules a plurality of movements. A random draw of one ruled movement moves a mover from one site to another. Players bet on selected movers moving by one or several rounds of drawing to selected sites. The holder of a hanging bet earns credit to place free make-up bets. Formulae to calculate the winning probability of every bet and its payoff and credit will be provided. An automatic computer/video version of the game is included.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a continuation of application Ser. No. 11/299,050, filed on Dec. 12, 2005, abandoned as this one is filed. This application is related to the application Ser. No. 691,944, filed on Aug. 5, 1996, now U.S. Pat. No. 5,795,226, granted on Aug. 18, 1998. The inventor's name was misprinted as Chen Yi. A certificate of correction was issued on Nov. 24, 1998.

BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to games of chance, more specifically, to methods of playing a betting game determined by one or multiple rounds of random numbers.

2. Prior art

As far as playing surface is concerned, every game with a plurality of moving pieces is prior art. As far as betting is concerned, any game of chance such as bingo, craps, keno, lottery or roulette is prior art. As far as technology is concerned, games requiring bet slips and computer data processing such as those at racetracks or slot/video games shown in patents listed in the Information Disclosure Statement are prior art.

3. Objects and Advantages

About the time of U.S. Pat. No. 5,795,226 being granted, I realized that the non-automatic version requires a 8′ by 8′ table, a rolling dice box, and so on, all made-to-order only. Its operation requires several workers. All this means high costs which will result in high house edge, something I hate. Why not replace the big table by a monitor display? Why not let a keno bowl of balls to generate random numbers? Why not allow players to determine own track length, and start a race anytime? etc. Besides, why not change the racing characteristic to movement from one site to another? Instead of finishing orders in one or more races, bet that Ann will make a trip from London to Paris followed by Bob from Beijing to Tokyo and then Sydney? So this invention originated.

The principal object of the invention is to provide a low operation cost multi-draw game of chance with all possible outcomes as well as their probabilities fully known to the public. It can be carried out by easily made to order equipment or existing keno/lottery facilities with minor changes. The bettor can easily mark a bet slip to place any amount of bets where the range of total winning probability lies from 99.9% to less than one billionth. It allows the operator to set a wide range of house edges so that players can either try to win big money with high negative expectation or enjoy gambling excitement with minimal negative expectation. Besides, there are bets to be determined by multiple rounds of random numbers so that make-up bets can be placed between two rounds. A hanging ticket holder can get credit to place free make-up bets and become a sure winner. This reduces the payoff of an original bet, but eliminates bettor's worry about bankroll and saves fund transaction handling. Furthermore, anywhere equipped with a TV or computer monitor connected to the game control center, anyone can place bets by pointer clicking or screen touching.

In early 1930s the Liberty Bell slot machines with 3-reel, 10 symbols per reel were installed in Las Vegas casinos for the purpose of keeping wives and girlfriends entertained while serious gamblers played at gaming tables. They became one-armed-bandits indicating that payoffs were very poor. Since 1990s they are the biggest moneymakers in casinos. Instead of 1000 possible stops, those mesmerizing monster devices with video displays and sound effects have unknown number of stops with unknown probabilities. The whole reel spinning process, mechanical or video, is under the control of a random number generator chip RNG which allows game manufacturers to design and customize the odds of each machine to ensure casinos the maximal takeout allowed by government regulations. Instead of one “pay-line”, there are now “multi-line”, “multi-stage”, “bonus-round”, “option-buy”, “scatter-pay”, “second-chance” etc., all just lures less intelligent people to bet more with an ever-present fantasy that jackpot will be bestowed on them. This invention is to provide a game for a huge group of potential players simply neglected by the gaming industry who never want to touch slots or video game machines with hidden random outcomes.

This invention is to provide a keno-like game without the following weaknesses of today's keno: 1. Too low winning probabilities and no low house edges. The highest keno winning probability is 1 in 4 while house edge is at least 25% (an exception will be given below). Most casino games have higher probabilities such as roulette “black-red”. House edge can be less than 1% such as craps “pass/come with 1× or higher odds”. Looking into statistic on annual casino revenue and state lottery ticket sale, one can see that the common gambling intention is inclined by far to catching more probable wins rather than becoming a millionaire fast. 2. Too troublesome to place desired amount of bets. There are 3.5 quintillion possible combinations of 20 out of 80. But, say, you want to play all possible 6-spot catch-all combinations with numbers 1, 2, 3 plus any three numbers from 4 to 80. No keno writer will be ready to help you mark at least a few hundred bet slips for your 73150 bets. If the spots of a group of numbers are not adjacent to each other, or if there are more than two groups to circle, confusion will likely occur in computerized digital scanning. 3. Every bet is determined by one draw. The only known exception is Exacta at Gold Coast, Las Vegas, which allows players to mark the same number of spots, from one to ten, in two consecutive games, paying $1 per game plus $0.25 for exacta. For the best payoff is thus to mark one spot in each game which pay $3 for first game, $3 for second game and $4 for exacta where house edges are 25%, 25% and 0% respectively. Unfortunately, due to 0% on 25 cents and 25% on $2, there is no way to take advantage of a hanging ticket by placing make-up bets after the first game. 4. No good method to generate random numbers. Keno uses a bowl of 80 whirling balls to push one at a time by air force into a selection tube until 20 are collected. The problem is, within a short period of action, not every ball can reach an appropriate position to be pushed. Besides, it occurred that a customer remarked to a keno manager at a Las Vegas casino that Number 29 never came up. Indeed, Number 29 was not in the cage. There will be no such problem if the game requires, say, to pick just one number out of seven while the cage contains 28 balls, four copies of each number.

Assume that at a racetrack your $10 Double ticket becomes hanging with a $500 “will pay” and 7 to 1 odds. Can you make yourself a sure winner? Due to over 20% take-out, only occasionally almost yes! For example, the occasion allows you to bet $200 on favorite with odds 6 to 5—payoff about $450—, and bet $150 on second favorite with odds 2 to 1—payoff about $450—, and bet $80 on third favorite with odds 5 to 1—payoff about $500—. Thus, you have two chances to earn $60, two chances $10, and as many chances as the number of all your neglected long shots to lose your total investment of $440. Regretfully in general, at a racetrack only hanging “Pick 3 or more” bet with “will pay” over one thousand dollars can result in a sure winner provided that your bank roll is ready to cover all required make-up bets. The game of invention is to put any hanging bet holder always in a position to become a sure winner by make-up bets scientifically.

Here is a very simple example of how to deal with a hanging bet scientifically. Say, you purchase a $100 2-Draw ticket on Ann to go to London first and then to Paris, each time one of seven possible destinations. When the first draw indeed moves Ann to London, you receive credit $700 for the hanging bet so that you can be a sure winner such as using 60% credit to bet $70 on each of six destinations other than Paris. The result will be a payoff of $40*49*94.13%=$1845.01, in case of Paris, or $70*7*94.13%=$461.25, in any other cases where house edges are all 5.87% based on probability 1/49. How about you don't want any credit? In order to be a sure winner, you need fund such as $3,600 to place $600 make-up bets on all six destinations other than Paris. The result will be a payoff of $100*49*94.13%=$4612.51 on your original ticket or $600*7*95.75%=$4025.10 on any one of make-up bets where house edge is 4.25% based on probability 1/7. This example clearly shows the game is fundamentally different such slot/video games as shown in U.S. Pat. No. 6,162,121, U.S. Pat. No. 6,855,052 etc. where the possible outcomes and their probabilities are programmed and unknown to the player so that there is simply no scientific way to place make-up bet.

Besides playing skillfully it can also be fun to hold a bet ticket stating such as “Dad, Mom and Me land on Moon”.

SUMMARY OF THE INVENTION

The invention provides a game of chance with a playing surface. The playing surface is a map of sites on which there are circled numbers called movers. A TV/computer monitor will be required to display the playing surface with movers.

The invention provides a plurality of ruled movements directing movers to move from one site to another. A random draw device functioning like the one used at keno will be required to draw ruled movements at random. Instead of 80 balls in one cage, here we use one cage for each mover in which there are equally many, say, four or five balls for each ruled movement.

The game requires a player called operator to conduct. Other players are bettors. The operator executes random draw of ruled movements, one round after another, moving each mover accordingly once per round. Henceforth every round of drawing will be simply called a draw.

The invention provides a plurality of bet slips showing either the whole playing surface with all movers or a part of the playing surface with a single mover. As at a racetrack bettors mark bet slips to place bets. The bettor can select one or several movers together with one or several sites which the bettor expects to match the outcomes of upcoming one or several draws.

Wagering machines connected to a computer with database wagering system software, similar to those used at a racetrack, will be required to examine marked bet slips, to print bet tickets showing all officially accepted bets, and to store and process betting data with drawing results. Regardless whether a draw is going on or not, the computer examines bet slips and issue bet tickets anytime such that no already started draw but the upcoming one will be effective. Bets can also be placed by means of wagering machines without paper bet slip.

A multi-Draw bet becomes or remains hanging if it contains a selection of movers with movements matching the last draw outcomes, and thus has a chance to be a winner later on. The invention provides the option that any hanging bet holder earns credit to place free make-up bets, henceforth called credit bets.

The winning probability of every bet as well as how to calculate payoffs and credits is included.

The invention provides a video/computer version of the game.

DRAWINGS

FIG. 1 is a flowchart illustrating the game process.

FIGS. 2 to 4 show similar playing surfaces.

FIG. 2A is a universal bet slip using playing surface as shown in FIG. 2.

FIGS. 3A, 4A are 1-Draw bet slips using playing surface as shown in FIGS. 3, 4 respectively.

FIGS. 3AA, 4AA are chain bet slips using playing surface as shown in FIGS. 3, 4 respectively.

FIGS. 2B, 3B and 4B are each a ‘simple’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.

FIGS. 2C, 3C and 4C are each a ‘site’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.

FIGS. 2D, 3D and 4D are each a ‘mixed’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.

FIG. 2E is a 4-Draw chain bet ticket using playing surface as shown in FIG. 2.

FIGS. 3E and 4E are each a 3-Draw chain bet ticket using playing surface as shown in FIGS. 3 and 4 respectively.

FIGS. 2F, 3F and 4F are each a Draw 2 revised chain bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.

FIGS. 2G, 3G and 4G are each a Draw 3 revised chain bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.

FIG. 2H is a Draw 4 revised chain bet ticket using playing surface as shown in FIG. 2.

FIG. 5 shows a betting activity statement.

FIG. 6 is a line graph to show house edge formula based on winning probability.

DESCRIPTION OF SIMILAR PLAYING SURFACES WITH RULED MOVEMENTS

Playing surfaces 10 in FIGS. 2, 3 and 4 will be shown on a monitor to indicate the locations of movers randomly set by the game operator before the start of the very first game or determined by the last draw of movements. Playing surfaces 100 in FIGS. 3AA, 4AA, etc. on bet slips are for players—optionally—to mark movers' locations as shown on monitor. Playing surfaces 101 in FIGS. 2A, 3A, etc. are for players to mark selections. Bet slips can be either printed or on-screen.

It is intended to enhance fun unrelated to game rules that on the bet slips movers will be given names of popular persons, and sites popular places.

A ruled movement aiming at a specific location will be called jump; otherwise, non-jump. As declared below, all movements on playing surface 10 as shown in FIG. 2 are jump, while those on playing surface 10 as shown in FIGS. 3 and 4 are non-jump. As defined below, for any mover, regardless of location, there is always the same number w of possible ruled movements.

Playing surface 10 in FIG. 2 contains seven sites 11 on which there are six movers 12.

There are w=7 ruled movements for this playing surface, denoted ‘A’, ‘B”, ‘C’, ‘D’, ‘E’, ‘F’, and ‘G’, as defined below:

-   -   ‘A’ moves the concerning mover to ‘A’, inclusive from ‘A’.     -   ‘B’ moves the concerning mover to ‘B’, inclusive from ‘B’.     -   ‘C’ moves the concerning mover to ‘C’, inclusive from ‘C’.     -   ‘D’ moves the concerning mover to ‘D’, inclusive from ‘D’.     -   ‘E’ moves the concerning mover to ‘E’, inclusive from ‘E’.     -   ‘F’ moves the concerning mover to ‘F”, inclusive from ‘F’.     -   ‘G’ moves the concerning mover to ‘G’, inclusive from G'.

Playing surface 10 in FIG. 3 contains ten sites 11 on which there are six movers 12. Similarly to most computer/video games, it is necessary to regard the top border line as identical to the bottom one. There are ‘A’ and ‘L’ painted outside playing surface 100 shown in FIGS. 3A and 3AA to visualizes this crossing border down/up situation. Thus, site ‘A’ lies one site downward to site ‘L’, two sites downward to site ‘K’; site ‘B’ lies two sites downward to site ‘L’; site ‘L’ lies one site upward to site ‘A’, two sites upward to site ‘B’ and three sites upward to site ‘C’; site ‘K’ lies two sites upward to site ‘A’ and three sites upward to site ‘B’; site ‘H’ lies three sites upward to site ‘A’.

There are w=6 ruled movements for this playing surface, denoted ‘00’, ‘U1’, ‘U2 ’, ‘U3’, ‘D1’, and ‘D2’, as defined below:

-   -   ‘00’ keeps the concerning mover unmoved.     -   ‘U1’ moves the concerning mover one site upward.     -   ‘U2’ moves the concerning mover two sites upward.     -   ‘U3’ moves the concerning mover three sites upward.     -   ‘D1’ moves the concerning mover one site downward.     -   ‘D2’ moves the concerning mover two sites downward.

Playing surface 10 in FIG. 4 contains twenty sites 11 on which there are six movers 12. Here it is necessary to regard the top border line as identical to the bottom one, the left border line identical to the right one. There are ‘DE’, ‘DA’, etc. painted outside playing surface 100 to visualizes this crossing border situation. Thus, site ‘AA’ lies surrounded by site ‘DA’ in the north, by site ‘DB’ in the northeast, by site ‘DE’ in the northwest, by site ‘AB’ in the east, by site ‘AE’ in the west, by site ‘BA’ in the south, by site ‘BB’ in the southeast, and by site ‘BE’ in the southwest; site ‘AB’ lies surrounded by site ‘DB’ in the north, by site ‘DC’ in the northeast, by site ‘DA’ in the northwest, by site ‘AC’ in the east, by site ‘AA’ in the west, by site ‘BB’ in the south, by site ‘BC’ in the southeast, and by site ‘BA’ in the southwest; and so on.

There are w=9 ruled movements for this playing surface, denoted ‘00’ N', ‘E’, ‘W’, ‘S’, ‘NE”, ‘NW’, ‘SE’, and ‘SW’, as defined below:

‘00’ keeps the concerning mover unmoved.

‘N’ moves the concerning mover to the adjacent site lying north.

‘E’ moves the concerning mover to the adjacent site lying east.

‘W’ moves the concerning mover to the adjacent site lying west.

‘S’ moves the concerning mover to the adjacent site lying south.

‘NE’ moves the concerning mover to the adjacent site lying northeast.

‘NW’ moves the concerning mover to the adjacent site lying northwest.

‘SE’ moves the concerning mover to the adjacent site lying southeast.

‘SW’ moves the concerning mover to the adjacent site lying southwest.

Description of Placing Bets

There are Draw 1 to Draw 4 1-Draw bets, further classified as ‘simple’, ‘site’ or ‘mixed’. There are multi-Draw or n-Draw ‘chain’ bets, where n=2 to 4, further classified as ‘linked’ or ‘unlinked’. All bets made on one bet slip are of the same class/type.

The bet slip as shown in FIG. 2A will be used for any bet concerning the playing surface as shown in FIG. 2. For playing surface as shown in FIG. 3, bet slip as shown in FIG. 3A will be used for 1-Draw bets while that of FIG. 3AA for chain ones. For playing surface as shown in FIG. 4, bet slip as shown in FIG. 4A will be used for 1-Draw bets while that of FIG. 4AA for chain bets. Besides the above regular bets using bet slips, there are, as explained later on, ‘credit’ bet using bet tickets. A mover once selected in a draw will be referred to as a bet-on mover of that draw. A site in which a bet-on mover is located will be referred to as selected site of that mover.

A linked chain bet slip will result in a single ticket and single winner. An unlinked chain bet slip will result in one ticket for each bet-on mover.

A common action on every bet slip is to mark a bet type, and either an ‘amount per bet’ or a ‘total bet amount’ except in the case of placing credit bet where ‘credit’ must be marked.

When movements are jump, all sites 11 in FIG. 2 are reachable by a single ruled movement and become sites 31, 41, 51 and 61 in playing surface 101 shown in FIG. 2A.

When movements are non-jump, referring to FIG. 3A or 4A, not every site 31 in playing surface 101 is reachable by a single ruled movement. Thus, any bet-on mover in an unreachable site will be simply cancelled by the computer when the bet slip is submitted for approval. By two or more movements, all sites 41, 51 and 61 are always reachable. Referring now to FIGS. 3AA and 4AA, instead of a complete playing surface, we use partial ones 101. In 101, where every mover before a draw is located in a gray site and all sites reachable by a single movement of Draw 1, Draw 2, Draw 3 and Draw 4 are sites 31, 41, 51 and 61 respectively.

In the following, * is the multiplication operator, and ̂ the exponent operator. Π(f(M)) is multiplication of f(M) over all M to be specified, and Σ(f(M)) is summation of f(M) over all M to be specified. Mathematically in general, M is a variable of function f, where f remains to be defined whenever needed.

To place 1-Draw bets, the bettor marks to select one or several movers 12 located in sites 31, 41, 51 and/or 61. Every selected 12 becomes a bet-on mover in a selected site. The bettor can play any one or more draws on the bet slip. All draws are independent. It is allowed, for example, to select some movers in sites 31, some in sites 51, but none in 41 or 61 for playing Draws 1 and 3.

Let #31(M), #41(M), #51(M) and #61(M) denote respectively the number of selected sites 31, 41, 51, and 61 of mover M.

In the ‘simple’ case, every selected site with a bet-on mover counts a bet. The same site will be counted as many times as the number of bet-on movers lying inside, draw by draw independently. Thus, the numbers of Draw 1, 2, 3, and 4 ‘simple’ bets are Σ(#31(M)), Σ(#41(M)), Σ(#51(M)), and Σ(#61(M)) respectively. The bettor wins, draw by draw independently, whenever there is a selected site 31 with one bet-on mover matching the outcomes of Draw 1, a selected site 41 with one bet-on mover matching the outcomes of Draw 2, a selected site 51 with one bet-on mover matching the outcomes of Draw 3, a selected site 61 with one bet-on mover matching the outcomes of Draw 4.

In the ‘site’ case, every selected site of all bet-on movers inside counts a bet. The bettor wins, draw by draw independently, whenever there is one selected site 31 with all bet-on movers inside matching the outcomes of Draw 1, one selected site 41 with all bet-on movers inside matching the outcomes of Draw 2, one selected site 51 with all bet-on movers inside matching the outcomes of Draw 3, one selected site 61 with all bet-on movers inside matching the outcomes of Draw 4.

In the ‘mixed’ case, every bet contains a complete set of all bet-on movers, each in connection with one selected site. For example, in Draw 1, there are three bet-on movers, one in 4 selected sites, another in 3 selected sites, and the third in 5 selected sites. Here are Π(#31(M))=4*3*5=60 bets. Thus, the numbers of Draw 1, 2, 3, and 4 ‘mixed’ bets are Π(#31(M)), Π(#41(M)), Π(#51(M)), and Π(#61(M)) respectively. The bettor wins, draw by draw independently, if, each time, for each one of all bet-on movers, there is one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, one selected site 51 matching the outcomes of Draw 3, or one selected site 61 matching the outcomes of Draw 4. The mixed bets on one ticket can bring in for each draw one winner only.

To place 2-Draw bets, the bettor marks to select first one or several movers in site 31 for Draw 1, and then one or several movers in sites 41 for Draw 2. The gray site 31 is where the mover located before Draw 1. Every Draw 1 bet-on mover must be bet-on in Draw 2 and vice versa. The gray site 41 is where the mover located after Draw 1—yet unknown—. So, for one and the same bet-on mover, any selected site 41 is valid for all selected sites 31, no matter what the outcomes of Draw 1 may be. If the bettor wants a certain selected site 41 just for a certain selected site 31, then it is necessary to use separate bet slips. —For example, using one bet slip you can bet a mover moves first either to east or west and then either to north or south. This makes four bets on one slip. If you want to bet that the mover moves either first to east then to north or first to west then to south, then you need to place them separately using two bet slips.—In the ‘unlinked’ case, the number of bets is Σ(#31(M)*#41(M)). —For example, there are bet-on movers A, B and C with #31(A)=4, #31(B)=3, #31(C)=5, #41(A)=2, #41(B)=6, and #41(C)=1, then Σ(#31(M)*#41(M))=(4*2)+(3*6)+(5*1).—The unlinked bets in one bet slip will result in one bet ticket for each bet-on mover. In the ‘linked’ case, the number of bets is Π(#31(M)*#41(M)). For example, there are bet-on movers A, B and C with #31(A)=4, #31(B)=3, #31(C)=5, #41(A)=2, #41(B)=6, and #41(C)=1, then Π(#31(M)*#41(M))=(4*2)*(3*6)*(5*1). A bet becomes hanging if for each one of all bet-on movers there is one selected site 31 matching the outcomes of Draw 1. Then it wins if for each one of all bet-on movers there is one selected site 41 matching the outcomes of Draw 2.

To place 3-Draw bets the bettor marks to select first just as explained in the 2-Draw case; then one or several movers in sites 51 for Draw 3. Every Draws 1 to 2 bet-on mover must be bet-on in Draw 3 and vice versa. The gray site 51 is where the mover located after Draw 2—yet unknown—. So, for one and the same bet-on mover, any selected site 51 is valid for all selected sites 31 and 41, no matter what the outcomes of Draws 1 and 2 may be. If the bettor wants a certain selected site 51 just for a certain selected sites 31 and 41, then it is necessary to use separate be slips. In the ‘unlinked’ case, the number of bets is Σ(#31(M)*#41(M)*#51(M)) which will result in one bet ticket for each bet-on mover. In the ‘linked’ case, the number of bets is Π(#31(M)*#41(M)*#51(M)). A bet becomes hanging if for each one of all bet-on movers there is one selected site 31 matching the outcomes of Draw 1. It remains hanging if for each one of all bet-on movers there is one selected site 41 matching the outcomes of Draw 2. Then it wins if for each one of all bet-on movers there is one selected site 51 matching the outcomes of Draw 3.

To place 4-Draw bets the bettor marks to select first just as explained in the 3-Draw case; then one or several movers in sites 61 for Draw 4. Every Draws 1 to 3 bet-on mover must be bet-on in Draw 4 and vice versa. The gray site 61 is where the mover located after Draw 3—yet unknown—. So, for one and the same bet-on mover, any selected Site 61 is valid for all selected sites 34, 44, and 51, no matter what the outcomes of Draws 1 to 3 draws may be. If the bettor wants a certain selected site 61 just for a certain selected sites 31, 41, and 51, then it is necessary to use separate bet slips. In the ‘unlinked’ case, the number of bets is Σ(#31 (M)*#41(M)*#51(M)*#61(M)) which will result in one bet ticket for each bet-on mover. In the ‘linked’ case, the number of bets is II (#31(M)*#41(M)*#51(M)*#61(M)). A bet becomes hanging if for each one of all bet-on movers there is one selected site 31 matching the outcomes of Draw 1. It remains hanging if for each one of all bet-on movers there is one selected site 41 matching the outcomes of Draw 2. It still remains hanging if for each one of all bet-on movers there is one selected site 51 matching the outcomes of Draw 3. Then it wins if for each one of all bet-on movers there is one selected site 61 matching the outcomes of Draw 4.

Every marked bet slip will be approved by the computer, in order to issue one or several bet tickets as shown in FIGS. 2B, 3B and 4B for simple bets, FIGS. 2C, 3C and 4C for site bets, FIGS. 2D, 3D and 4D for mixed bets, and FIGS. 2E, 3E and 4E for chain bets. Each ticket shows type of bets, per bet amount (if marked by the bettor), total number of bets, total bet amounts, and the Draw # of Draw 1. A playing surface with start locations of all movers before Draw 1 will be printed unless the ruled movements are jumps. In the case of 1-Draw bets, only bet-on movers in selected sites will show up on the bet ticket. In the case of chain bets, bet-on movers in selected sites will be marked with “X”. It is for the sake of convenience to allow unlinked bets on one slip. In order to avoid confusion and to make credit bets simple, there will be no unlinked chain bet ticket. Any unlinked bet slip of n bet-on movers will result in issuing n chain bet tickets, one for each bet-on mover. Thus, every chain bet ticket with more than one bet-on mover is ‘linked’.

After Draw 1, a hanging ‘chain’ ticket can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not.—Calculation of credit amount will be explained later on—What a bettor has to do is to select a certain percentage point. In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. In the case of no new bet slip, every credit bet is a 1-Draw mixed bet without any new bet-on mover, the player marks to select for each bet-on mover one or several sites 41. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #41′(M) denote the number of both originally and newly selected sites 41 for mover M. The number of credit bets will be #41(cr)=Π(#41′(M))−Π(#41(M)). This ticket as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2F, 3F or 4F can be issued. The revised ticket shows all original data. It also shows selected credit percentage, and all new selections marked with “=”. Besides, each mover located in the site determined by Draw 1 will be printed in gray. All credit bets marked on a hanging ticket are Draw 2 ‘mixed’, or ‘simple’ if there is only one bet-on mover.

After Draw 2 a hanging ‘chain’ ticket—whether revised or not—can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not. In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘Credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. The revised ticket remains hanging if one selection matches the outcomes of Draw 3. In the case of no new bet slip, every credit bet is a 1-Draw mixed bet without any new bet-on mover, the player marks to select for each bet-on mover one or several sites 51. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #51′(M) denote the number of both originally and newly selected sites 51 for mover M. The number of credit bets will be #51(cr)=Π(#51′(M))−Π(#51(M)). This ticket used as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2G, 3G or 4G can be issued. The revised ticket shows all original data. It also shows selected credit percentage, and all new selections marked with “=”. Besides, each mover located in the site determined by Draw 2 will be printed in gray. All credit bets marked on a hanging ticket are Draw 3 ‘mixed’ or ‘simple’ if there is only one bet-on mover.

After Draw 3 a hanging chain ticket—whether revised or not—can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not. In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘Credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. In the case of no new bet slip, every credit bet is a 1-Draw mixed bet without any new bet-on mover, the player marks to select for each bet-on mover one or several sites 61. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #61′(M) denote the number of both originally and newly selected sites 61 for mover M. The number of credit bets will be #61(cr)=Π(#61′(M))−Π(#61(M)). This ticket as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2H can be issued. The revised ticket shows all original data. It also shows selected credit percentage, and all new selections marked with “=”. Besides, each mover located in the site determined by Draw 3 will be printed in gray. All credit bets marked on a hanging ticket are Draw 4 ‘mixed’ or ‘simple’ if there is only one bet-on mover.

Random Number Generator

There will be one manipulation-proof random number generator for each mover to pick ruled movements. It can be a mechanical device like the one used at keno. While there a number on each ball, here a symbol such as ‘C’, ‘U2 ’ or ‘NE’ representing one ruled movement. While there one cage with 80 balls, here one cage with equally many, say, four or five balls for each movement. The generator can also be partly electronic using so-called computer TIMER function with 8,640,000 centi-seconds per day so that every centi-second is assigned to one movement such as it will be movement ‘X’ when 3,456,789 centi-seconds have elapsed since midnight. Which centi-second is assigned to which movement can be made known to the public. INT(8,640,000/w) centi-seconds will be assigned to reach one of w given distinct movements. Note: INT(x) is x if x is a whole number; otherwise the largest whole number less than x. [8,640,000−w*INT(8,640,000/w)] seconds are assigned to ‘void’. There is no fear of manipulation because pressing a button to draw by a finger nobody is able to catch one desired elapsed centi-second of a day. Anyway, the random number generator shall obviously produce all ruled movements equally probable at random.

Description of the Non-Automatic Game

The game requires a player, called operator, to start by putting movers 12 in sites 11 arbitrarily as shown in FIGS. 2, 3 and 4. All other players, called bettors, use paper or on-screen slip as shown in FIGS. 2A, 3A, 3AA, 4A and 4AA to place bets as described above. At a preset time, independent of wagering activity, the operator uses random generators described above to execute the first round of random drawing of ruled movements, called Draw #1, one movement for each mover. The outcomes will be displayed on the monitor, and input into the computer to determine if any bet ticket contains selections matching the outcomes so that its holder can obtain payoff or credit.

Whether there is a drawing in action or not, whether having placed bets before Draw #1 or not, any bettor can place bets anytime just like before Draw #1. Besides, it is an option that the hanging ticket holder can place credit bets. At a preset time the operator executes the next round of drawing, called Draw #2, for all movers. The outcomes will be displayed and data processed just like after Draw #1. As the flowchart in FIG. 1 shows, the above steps repeat. Unless pause or stop has been regulated ahead, the operator will let it go on indefinitely, while any player may start to bet or stop betting anytime. The Draw # will grow accordingly. But players don't need to pay attention to it. For the sake of reference, ‘Draw 1 is Draw # so and so’ will be printed on every bet ticket. A regulated stop must allow any existing bets to reach final results.

Naturally, anywhere equipped with a TV/computer monitor connected to the game control center, anyone can place bets by pointer clicking or screen touching.

Description of the Automatic Version

To play the automatic game, one needs either a video game machine or a personal computer equipped with made-to-order software inclusive TIMER function random number generator to take care of drawing ruled movements. The computer is connected to a pointing device or touch screen monitor so that the action ‘select’ below can be executed by means of the pointer clicking or finger touching. Selecting any item on the display screen will either highlight it or result in a new display. Selecting a highlighted item is to cancel that selection. In the non-automatic game, all figures printed on paper are supposed to be black, white and gray. On monitor they are colorful.

The game starts with the display of a playing surface as shown in FIG. 2, 3 or 4 with additional icons/items named “Another playing surface”, “Bet slip” and “Account”.

Selecting “Another playing surface” will result in the display of another one. All playing surfaces as shown in FIGS. 2 to 4, or maybe some one not given here, will be displayed cyclically one after another if the selection of “Another playing surface” continues.

Selecting “Bet slip” will display a bet slip as shown in FIG. 2A, 3A, 3AA, 4A or 4AA with additional icons “Ticket” and “Account”; and “Alternative slip” if playing surface as shown in FIG. 3 or 4 is used.

Selecting “Alternative slip” will switch to a chain bet slip if the displayed one is for 1-Draw bet, or conversely.

The player places bets on screen just as on paper in the non-automatic game; then selects “ticket” to submit. If the submitted slip is incomplete or contains error, there will be a message like ‘Incomplete! Please select bet amount’, requiring the player to make amendment. If the submission is approved, a bet ticket without a ticket number as shown in FIGS. 2B to 2D, 3B to 3D or 4B to 4D with additional icons “Go back”, “Ticket #”, “Cancel”, “Bet slip”, “Draw”, “Account” shows up.

Facing a bet ticket the player must select “Ticket #”, “Go back” or “Cancel”. Otherwise, there will be a message to remind the player to do so. Selecting “Ticket#” finalizes the bet so that a certain ticket number will be issued and shown on the ticket. Selecting “Go back” allows the player make changes on the submitted bet slip. Selecting “Cancel” is to abandon the submission.

After ‘Ticket # so and so’ or ‘Cancelled’ being displayed, the player can select “Bet slip”, “Draw”, or “Account”.

Selecting “Bet slip” will display a blank one to take bet.

Selecting “Draw” will cause one round of drawing followed by an outcome display as shown in FIGS. 2 to 4 with an additional icon “Account”. At the same time, the computer will update and process all bet ticket data.

Selecting “Account’ will result in a display as shown in FIG. 5. It shows available balance and all betting activities since the start of the game or the opening of that account. There is one account for each playing surface.

Here the player can select “Ticket # so and so” to view that ticket as well as to use it for placing credit bets just as in the non-automatic game.

“Playing surface”, “Bet slip” or “Draw” allows the player to continue in whichever way preferred, while “Exit” to end the game.

Probabilities

In the case of 1-Draw bet using bet slip as shown in FIG. 2A, any ‘simple’ bet has winning probability p=1/w. Any ‘site’ bet has winning probability p=1/ŵm where m is the number of bet-on movers in that site of the concerning draw. Any ‘mixed’ bet has winning probability p=1/ŵm where m is the number of bet-on movers of the concerning draw.

In the case of 1-Draw bet using bet slip as shown in FIG. 3A or 4A we need to specify a bet by the relative start-to-end locations explained below.

Referring to FIG. 3A, the identification of top with bottom border lines allows us to assign any one of the ten sites with 1-dimensional coordinates x and others with coordinates x+i, where every calculation involving i or x is modulo 10 arithmetic. Now we can replace A, B, etc by x, x+1, etc. respectively. —For example, A:0, B:9, C:8, D:7, E:6, F:5, G:4, H:3, K:2, L:1.—And we can also say that x+i lies i sites away from x. A movement from 0 to i is equivalent to a movement from x to x+i. Thus, 1-movement path d1(i) are as follows:

-   -   Movement ‘00’ moves a mover from x to x, defining a 1-movement         path d1(0),     -   Movement ‘U1’ moves a mover from x to x+1, defining a 1-movement         path d1(1),     -   Movement ‘U2’ moves a mover from x to x+2, defining a 1-movement         path d1(2),     -   Movement ‘U3’ moves a mover from x to x+3, defining a 1-movement         path d1(3),     -   Movement ‘D1’ moves a mover from x to x+9, defining a 1-movement         path d1(9),     -   Movement ‘D2’ moves a mover from x to x+8, defining a 1-movement         path d1(8).

Let #d1(i) denote the number of all d1(i) for i. Obviously we have #d1(0)=#d1(1)=#d1(2)=#d1(3)=#d1(8)=#d1(9)=1 and #d1(4)=#d1(5)=#d1(6)=#d1(7)=0; in total 6.

p1M=#d1(i(M))/w is the probability of mover M from its start location to get on a d1(i(M)) path to reach the site 31 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.

Let d2(i) be any d1(x) followed by any d1(i−x), defining a 2-movement path from any site to a site lying i sites away.

Let #d2(i) denote the number of all d2(i) for i. #d2(i) is the sum of #d1(x)*#d1(i−x) over all x; explicitly, we have #d2(0)=5, #d2(1)=6, #d2(2)=5, #d2(3)=4, #d2(4)=3, #d2(5)=2, #d2(6)=2, #d2(7)=2, #d2(8)=3, #d2(9)=4; in total 36, that is 6̂2.

p2M=#d2(i(M))/ŵ2 is the probability of mover M from its start location to get on a d2(i(M)) path to reach the site 41 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.

Let d3(i) be any d1(x) followed by any d2(i−x), defining a 3-movement path from any site to a site lying i sites away.

Let #d3(i) denote the number of all d3(i) for i. #d3(i) is the sum of #d1(x)*#d2(i−x) over all x; explicitly, we have #d3(0)=25, #d3(1)=27, #d3(2)=27, #d3(3)=25, #d3(4)=22, #d3(5)=18, #d3(6)=16, #d3(7)-16, #d3(8)=18, #d2(9)=22; in total 216, that is 6̂3.

p3M=#d3(i(M))/ŵ3 is the probability of mover M from its start location to get on a d3(i(M)) path to reach the site 51 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.

Let d4(i) be any d1(x) followed by any d3(i−x), defining a 4-movement path from any site to a site lying i sites away.

Let #d4(i) denote the number of all d4(i) for i. #d4(i) is the sum of #d1(x)*#d3(i−x) over all x; explicitly, we have #d4(0)=135, #d4(1)=144, #d4(2)=148, #d4(3)=144, #d4(4)=135, #d4(5)=124, #d4(6)=115, #d4(7)=112, #d4(8)=115, #d4(9)=124; in total 1296, that is 6̂4.

p4M=#d4(i(M))/ŵ4 is the probability of mover M from its start location to get on a d4(i(M)) path to reach the site 61 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.

A Draw n simple bet, where n=1 to 4, on mover M lying i(M) sites away will be denoted by dn(i(M)). It has winning probability p=pnM=#dn(i(M))/ŵn.

A Draw n site bet, where n=1 to 4, on movers M lying each i(M) sites away from site S will be denoted by dnS( . . . , i(?), . . . ), where ? goes from mover #1 to #6, and i(?) is i(M) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d3B(-,2,1,-,-,8) is a Draw 3 site bet on site B with bet-on movers #2, #3 and #6, lying respectively 2, 1 and 8 sites away from site B. Or, d4E(3,2,1,-,7,-) is a Draw 4 site bet on site E with bet-on movers #1, #2, #3 and #5, lying respectively 3, 2, 1 and 7 sites away from site E. The dnS( . . . , i(?), . . . ) bet has the winning probability of p=Π(pnM) where multiplication is over bet-on movers in site S of Draw n.

A Draw n mixed bet, where n=1 to 4, on movers M lying each i(M) sites away will be denoted by dnX( . . . , i(?), . . . ), where ? goes from mover #1 to #6, and i(?) is i(M) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d2X(2,5,-,-,-,6) is a Draw 2 mixed bet with bet-on movers #1, #2 and #6, lying respectively 2, 5 and 6 sites away. Or, d3X(-,-,3,2,3,-) is a Draw 3 mixed bet with bet-on movers #3, #4 and #5, lying respectively 3, 2 and 3 sites away. The dnS( . . . , i(?), . . . ) bet has the winning probability of p=Π(pnM) where multiplication is over all bet-on movers of Draw n.

Referring now to FIG. 4A, the identification of top with bottom border lines and left with right border lines allows us to assign any one of the twenty sites with matrix coordinates (x,y) and others with coordinates (x+i,y+j), where every calculation involving i or x is modulo 4 arithmetic, involving j or y is modulo 5 arithmetic. Now we can replace AA, AB, etc. by (x,y), (x,y+1) etc. respectively.—For example, AA:(0,0), AB:(0,1), AC:(0,2), AD:(0,3), AE:(0,4), BA:(1,0), BB:(1,1), BC:(1,2), BD:(1,3), BE:(1,4), CA:(2,0), CB:(2,1), CC:(2,2), CD:(2,3), CE:(2,4), DA:(3,0), DB:(3,1), DC:(3,2), DD:(3,3), DE:(3,4).—And we can also say that (x+i,y+j) lies (i,j) sites away from (x,y). A movement from (0,0) to (i,j) is equivalent to a movement from (x,y) to (x+i,y+j). Thus, 1-movement path d1(i,j) are as follows:

‘00’ moves a mover from (x,y) to (x,y), defining a 1-movement path d1(0,0).

‘E’ moves a mover from (x,y) to (x,y+1), defining a 1-movement path d1(0,1)

‘W’ moves a mover from (x,y) to (x,y+4), defining a 1-movement path d1(0,4)

‘N’ moves a mover from (x,y) to (x+3,y), defining a 1-movement path d1(3,0).

‘S’ moves a mover from (x,y) to (x+1,y), defining a 1-movement path d1(1,0).

‘NE’ moves a mover from (x,y) to (x+3,y+1), defining a 1-movement path d1(3,1).

‘SE’ moves a mover from (x,y) to (x+1,y+1), defining a 1-movement path d1(1,1).

NW' moves a mover from (x,y) to (x+3,y+4), defining a 1-movement path d1(3,4).

‘SW’ moves a mover from (x,y) to (x+1,y+4), defining a 1-movement path d1(1,4).

Let #d1(i,j) denote the number of all d1(i,j) paths from (0,0) to (i,j). Obviously, we have #d1(0,0)=#d1(0,1)=#d1(0,4)=#d1(1,0)=#d1(1,1)=#d1(1,4)=#d1(3,0)=#d1(3,1)=#d1(3,4)=1 and #d1(i,j)=0 for all other (i,j); in total 9.

p1M=#d1(i(M),j(M))/w is the probability of mover M from its start location to get on a d1(i(M),j(M)) path to reach the site 31 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.

Let #d2(i,j) denote the number of all d1(i,j) paths from (0,0) to (i,j). #d2(i,j) is the sum of #d1(p,q)*#d1(i−p,j−q) over all p and p; explicitly, we have #d2(0,0)=9, #d2(0,1)=6, #d2(0,2)=3, #d2(0,3)=3, #d2(0,4)=6, #d2(1,0)=6, #d2(1,1)=4, #d2(1,2)=2, #d2(1,3)=2, #d2(1,4)=4, #d2(2,0)=6, #d2(2,1)=4, #d2(2,2)=2, #d2(2,3)=2, #d2(2,4)=4, #d2(3,0)=6, #d2(3,1)=4, #d2(3,2)=2, #d2(3,3)=2, #d2(3,4)=4; in total 81, that is 9̂2.

p2M=#d2(i(M),j(M))/ŵ2 is the probability of mover M from its start location to get on a d2(i(M),j(M)) path to reach the site 41 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.

Let d3(i,j) be any d1(x,y) followed by any d2(i−x,j−y), defining a 3-movement path from any site to a site lying (i,j) sites away.

Let #d3(i,j) denote the number of all d3(i,j) paths from (0,0) to (i,j). #d3(i,j) is the sum of #d1(p,q)*#d2(i−p,j−q) over all p and p; explicitly, we have #d3(0,0)=49, #d3(0,1)=42, #d3(0,2)=28, #d3(0,3)=28, #d3(0,4)=42, #d3(1,0)=49, #d3(1,1)=42, #d3(1,2)=28, #d3(1,3)=28, #d3(1,4)=42, #d3(2,0)=42, #d3(2,1)=36, #d3(2,2)=24, #d3(2,3)=24, #d3(2,4)=36, #d3(3,0)=49, #d3(3,1)=42, #d3(3,2)=28, #d3(3,3)=28, #d3(3,4)=42, in total 729, that is 9̂3.

p3M=#d3(i(M),j(M))/ŵ3 is the probability of mover M from its start location to get on a d3(i(M),j(M)) path to reach the site 51 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.

Let d4(i,j) be any d1(x,y) followed by any d3(i−x,j−y), defining a 4-movement path from any site to a site lying (i,j) sites away.

Let #d4(i,j) denote the number of all d4(i,j) paths from (0,0) to (i,j). #d4(i,j) is the sum of #d1(p,q)*#d3(i−p,j−q) over all p and q; explicitly, we have #d4(0,0)=399, #d4(0,1)=357, #d4(0,2)=294, #d4(0,3)=294, #d4(0,4)=357, #d4(1,0)=380, #d4(1,1)=340, #d4(1,2)=280, #d4(1,3)=280, #d4(1,4)=340, #d4(2,0)=380, #d4(2,1)=340, #d4(2,2)=280, #d4(2,3)=280, #d4(2,4)=340, #d4(3,0)=380, #d4(3,1)=340, #d4(3,2)=280, #d4(3,3)=280, #d4(3,4)=340; in total 6561, that is 9̂4.

p4M=#d4(i(M),j(M))/ŵ4 is the probability of mover M from its start location to get on a d4(i(M),j(M)) path to reach the site 61 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.

A Draw n simple bet, where n=1 to 4, on mover M lying each (i(M),j(M)) sites away will be denoted by dn((i(M),j(M)). It has winning probability p=pnM.

A Draw n site bet, where n=1 to 4, on movers M lying each (i(M),j(M)) sites away from site S will be denoted by dnS( . . . , (i(?),j(?)), . . . ), where ? goes for each (i,j) from mover #1 to #6, and (i(?),j(?)) is (i(M),j(M)) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d3BB(-,(2,3), (3,1),-,-,(0,4)) is a Draw 3 site bet on site BB with bet-on movers #2, #3 and #6, lying respectively (2,3),(3,1) and (0,4) sites away from site BB. Or, d4DA((3,3),(0,2),(1,1),-,(0,0),-) is a Draw 4 site bet on site DA with bet-on movers #1, #2, #3 and #5, lying respectively (3,3),(0,2),(1,1) and (0,0) sites away from site DA. The dnS( . . . , (i(?), j(?)), . . . ) bet has winning probability p=Π(pnM) where multiplication is over all bet-on movers in site S of Draw n.

A Draw n mixed bet, where n=1 to 4, on movers M lying each (i(M),j(M)) sites away will be denoted by dnX( . . . ,(i(?),j(?)), . . . ), where ? goes for each (i,j) from mover #1 to #6, and (i(?),j(?)) is (i(M),j(M)) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d2X((3,2),(2,3),-,-,-, (1,1)) is a Draw 2 mixed bet with bet-on movers #1, #2 and #6, lying respectively (3,2),(2,3) and (1,1) sites away. Or, d3X(-,-,(0,3),(1,2),(3,4),-) is a Draw 3 mixed bet with bet-on movers #3, #4 and #5, lying respectively (0,3),(1,2) and (3,4) sites away. The dnX( . . . ,(i(?),j(?)), . . . ) bet has winning probability p=Π(pnM) where multiplication is over all bet-on movers of Draw n.

For all playing surfaces:

A 2-Draw chain bet with m bet-on movers has probability 1/ŵm to become hanging, and probability 1/ŵ2m to win.

A 3-Draw chain bet with m bet-on movers has probability 1/ŵm to become hanging, and probability 1/ŵ2m to remain hanging, and probability 1/ŵ3m to win.

A 4-Draw chain bet with m bet-on movers has probability 1/ŵm to become hanging, and probability 1/ŵ2m to remain hanging, and probability 1/ŵ3m to remain hanging again, and probability 1/ŵ4m to win.

House Edges, Payoffs and Credits

This game requires reasonable house edges such as follows: Let x be the inverse of the product of winning probabilities of all involved draws. (see FIG. 6)

e(x)=2.5+x/4 for 0<x≦10

e(x)=4+(n+1)[n/2+(x−10̂n)/90(10̂n)] for 10<x with integer n satisfying 10̂n<x≦10̂(n+1).

Let p1 to p4 be the winning probabilities of Draw 1 to 4 respectively.

Let r2, r3 and r4 be the percentage points of credit selected by the bettor for Draw 2, 3 and 4 respectively.

A $a 1-Draw bet pays $a*(100−e(1/p1))%/p1.

A $a 2-Draw bet pays $a*(100−r2)%*(100−e(1/(p1*p2)))%/(p1*p2).

A $a 3-Draw bet pays $a*(100−r2)%(100−r3)%*(100−e(1/(p1*p2*p3)))%/(p1*p2*p3).

A $a 4-Draw bet pays $a*(100−r2)%*(100−r3)%*(100−r4)%*(100−e(1/(p1*p2*p3*p4)))%/(p1*p2*p3*p4).

A $a n-Draw bet hanging after Draw 1 earns credit $a/p1.

A $a n-Draw bet hanging after Draw 2 earns credit $a*(100−r2)%/(p1*p2).

A $a n-Draw bet hanging after Draw 3 earns credit $a*(100−r2)%*(100−r3)%/(p1*p2*p3).

A $b credit bet placed after Draw 1 pays $b*(100−e(1/(p1*p2)))%/p2.

A $c credit bet placed after Draw 2 pays $c*(100−e(1/(p1*p2*p3)))%/p3.

A $d credit bet placed after Draw 3 pays $d*(100−e(1/(p1*p2*p3*p4)))%/p4.

Any credit bet placed using a new slip bears a carryover inverse y of the product of all winning probabilities related to the bets resulting in credit so that in the payoff calculation house edge e will be a function of y*z, where z is the inverse of the product of all winning probabilities related to bets on the new ticket.

Numerical Examples

In order to make calculations less complex, no house edge will be applied below:

The ‘simple’ bet ticket as shown in FIG. 2B has in Draw 1 the probability of 2/7, 3/7, 2/7, 1/7, 5/7, 3/7 respectively to win $7 each on movers #1 to #6; in Draw 2 the probability of 5/7, 2/7, 2/7, 2/7, 3/7, 1/7 respectively to win $7 each on movers #1 to #6; in Draw 3 the probability of 2/7, 2/7, 3/7, 3/7, 3/7, 2/7 respectively to win $7 each on movers #1 to #6; and in Draw 4 the probability of 4/7, 1/7, 2/7, 2/7, 3/7, 2/7 respectively to win $7 each on movers #1 to #6.

The ‘simple’ ticket as shown in FIG. 3B has in Draw 1 the probability of 4/6, 3/6, 2/6, 5/6, 2/6, 3/6 respectively to win $12 each on movers #1 to #6. In Draw n, where n=2 to 4, every dn(i(M)) bet has probability p=#dn(i(M))/ŵn to win payoff $2/p. For example, d2(8(3)) bet has p=3/6̂2; d3(4(4)) bet has p=22/6̂3; d4(2(2)) bet has p=148/6̂4.

The ‘simple’ ticket as shown in FIG. 4B has in Draw 1 the probability of 7/9, 0/9, 8/9, 6/9, 2/9, 4/9 respectively to win $9 each on movers #1 to #6. In Draw n, where n=2 to 4, every dn(i(M),j(M)) bet has probability p=#dn(i(M),j(M))/ŵn to win payoff $1/p. For example, d2(0(2),1(2)) bet has p=6/9̂2; d3(2(5),1(5)) bet has p=36/9̂3; d4(3(6),3(6)) bet has p=280/9̂4.

Every one of 25 ‘site’ bets in the ticket as shown in FIG. 2C has winning probability p=1/7̂m, where m is the number of bet-on movers in the selected site, to win payoff $2/p.

In the ‘site’ ticket as shown in FIG. 3C, every dnS( . . . , i(M), . . . ) bet, where n=1 to 4, has probability p=Π(pnM), where multiplication is over all bet-on movers of Draw n, to win payoff $2/p. For example, d1C(1,-,-,-,0,-) bet has p=#d1(1)*#d1(0)/6̂2=1/6̂2; d2H(6,0,8,3,0,0) bet has p=#d2(6)*#d2(8)*#d2(3)/6̂6=2*3*4/6̂3; d3G(7,1,9,4,-,-) bet has p=#d3(7)*#d3(1)*#d3(9)*#d3(4)/6̂4=16*27*22*25/6̂4; d4F(8,-,0,5,7,9,0) bet has p=#d4(8)*#d4(0)*#d4(5)*#d4(9)* #d4(0)/6̂5=115*135*112*124*135/6̂5.

In the ‘site’ ticket as shown in FIG. 4C, every dnS( . . . , (i(M),j(M)), . . . ) bet, where n=1 to 4, has probability p=Π(pnM), where multiplication is over all bet-on movers in site S, to win payoff $2/p. For example, d1BB(-,-,-,(3,1),(1,0),-) bet has p=#d1(3,1)*#d1(1,0)/9̂2=1/9̂2; d2CB(-,-,(1,2),-, (2,0),-) bet has p=#d2(1,2)*#d2(2,0)/9̂4=2*6/9̂4; d3BA((1,2),-,-,-,-,(2,2)) bet has p=#d3(1,2)*#d3(2,2)/9̂6=28*24/9̂6; d4DE((3,1),-,(2,0),-,-,(0,1)) bet has p=#d4(3,1)*#d4(2,0)*#d4(0,1)/9̂12=340*380*357/9̂12.

Every one of 622 ‘mixed’ bets in the ticket as shown in FIG. 2D has probability p=1/7̂m, where m is the number of bet-on movers of the concerning draw, to win payoff ($100/622)/p.

In the ‘mixed’ ticket as shown in FIG. 3D every dnX( . . . , i(M), . . . ) bet, where n=1 to 4, has probability p=Π(pnM), where multiplication is over all bet-on movers of the concerning draw, to win payoff $0,10/p. For example, d1X(0,3,1,3,8,2) bet has p=#d1(0)*#d1(3)*#d1(1)*#d1(3)*#d1(8)*#d1(2)/6̂6=1/6̂6; d2X(1,4,8,2,7,2) bet has p=#d2(1)*#d2(4)*#d2(8)*#d2(2)*#d2(7)*#d2(2)/6̂12=6*3*5*2*5/6̂12; d3X(1,3,8,4,9,9) bet has p=#d3(1)*#d3(3)*#d3(8)*#d3(4)* #d3(9)*#d3(9)/6̂18=27*25*18*22*22*22/6̂18; d4X(8,2,9,3,0,2) bet has p=#d4(8)*#d4(2)*#d4(9)*#d4(3)*#d4(0)*#d4(2)/6 ̂24=115*148*124*144*135*148/6̂24.

In the ‘mixed’ ticket as shown in FIG. 4D every dnX( . . . , (i(M), j(M)), . . . ) bet, where n=1 to 4, has probability p=Π(pnM), where multiplication is over all bet-on movers of that draw, to win payoff $0,10/p. For example, d1X((0,4),-,(3,4),(3,0),(1,0),(3,0)) bet has p=#d1(0,4)*#d1(3,4)*#d1(3,0)*#d1(1,0)*#d1(3,0)/9̂5=1/9″5. d2X(-,(3,4),(1,3),(0,0),(0,1),-) bet has p=#d2(3,4)*#d2 (1,3)* #d2(0,0)*#d2(0,1)/9̂8=4*2*9*6/9̂8; d3X((0,2),(3,4),-,-,(1,0),(0,0)) bet has p=#d3(0,2)*#(13(3,4)*#d3(1,0)*#d3(0,0)/9̂12=28*42*49*49/9̂12; d4X((1,2),(2,4),(1,4),(3,2),-, (3,0)) bet has p=#d4(1,2)*#d4(2,4)*#d4(1,4)*#d4(3,2)*#d4(3,0)/9̂20=280*340*340*280*380/9̂20.

The 4-Draw ticket as shown in FIG. 2E has probability p=p1*p2*p3*p4 where p1=(4*4*5*5/7̂4)=400/2401, p2=3*2*4*4/7̂4=96/2401, p3=3*4*3*3/7̂4=108/2401, and p4=2*2*2*2/7̂4=16/2401 to win payoff $200/p=$100,166,770.86. It also has probability p1 to become hanging and earn Draw 2 credit $200/p1=$1,200.50.

The revised 4-Draw ticket as shown in FIG. 2F with #41(cr)=1008-96 credit 70% bets has probability q2=#41(cr)/7̂4=912/2401 to win payoff $200*0.01*70/(p1*q2)=$2212.37. It also has probability p2 to earn Draw 3 credit $200*0.01*(100-70)/(p1*p2)=$9,007.50.

The revised 4-Draw ticket as shown in FIG. 2G with #51(cr)=1512−108 60% credit 70% bets has probability q3=#51(cr)/7̂4=1404/2401 to win payoff $200*0.01*(100−70)*0.01*60/(p1*p2*q3)=$9,242.31. It also has probability p3 to earn Draw 4 credit $200*0.01*(100-70)* 0.01*(100-60)/(p1*p2*p3)=$80,100.04.

The revised 4-Draw ticket as shown in FIG. 2H with #61(cr)=2401-16 credit 80% bets has probability q4=#61(cr)/7̂4=2385/2401 to win payoff $200*0.01*(100-70)*0.01*(100-60)*0.01*80/(p1*p2*p3*q4)=$64,509.92 on one credit bet. It also has probability p4 to win payoff $200*0.01*(100−70)*0.01*(100−60)*0.01*(100-80)/(p1*p2*p3*p4)=$2,404,002.60 on one original bet.

The 3-Draw ticket as shown in FIG. 3E has probability p=p1*p2*p3 where p1=5*5*5*5*5/6̂5=3125/7776 p2=4*4*4*4*4/6̂5=1024/7776, and p3=3*2*3*2*3/6̂5=108/7776 to win payoff $200/(p1*p2*p3)=$272,097.79. It also has probability p1 to earn Draw 2 credit $200*p1=$497.66, The revised 3-Draw ticket as shown in FIG. 3F with #41(cr)=7776-1024 credit 70% bets has probability q2=#41(cr)/6̂5=6752/7776 to win payoff $200*0.01*70/(p1*q2)=$401.20. It also has probability p2 to earn Draw 3 credit $200*0.01*(100-70)/(p1*p2)=$113.37.

The revised 3-Draw ticket as shown in FIG. 3G with #51(cr)=7776-108 credit 80% bets has probability q3=#51(cr)/6̂5=7668/7776 to win payoff $200*0.01*(100−70)*0.01*80/(p1*p2*q3)=$919.77 on one credit bet. It also has probability p3 to win payoff $200*0.01*(100-70)*0.01*(100−80)/(p1*p2*p3)=$16,325.87 on one original bet.

The 3-Draw ticket as shown in FIG. 4E has probability p=p1*p2*p3 where p1=8*8*8*7/9̂4=4704/6561, p2=7*7*6*6/9̂4=1764/6561 and p3=6*5*5*5/9̂4=750/6561, to win payoff $200/(p1*p2*p3)=$9,076.39. It also has probability p1 to earn Draw 2 credit $200/p1=$278.95.

The revised 3-Draw ticket as shown in FIG. 4F with #41(cr)=5832-1764 credit 60% bets has probability q2=#41(cr)/9̂4=4068/6561 to win payoff $200*0.01*60/(p1*q2)=$269.94. It also has probability p2 to earn Draw 3 credit $200*0.01*(100-60)/(p1*p2)=$415.02.

The revised 3-Draw ticket as shown in FIG. 4G with #51(cr)=5184-750 credit 70% bets has probability q3=#51(cr)/9̂4=4434/6561 to win payoff $200*0.01*(100-60)*0.01*70/(p1*p2*q3)=$429.87 on one credit bet. It also has probability p3 to win payoff $200*0.01*(100-60)*0.01*(100−70)/(p1*p2*p3)=$1,089.17 on one original bet.

CONCLUSION

The invention described above provides an extremely low operation cost game to be easily run by an existing or future keno/lottery kind of operator.

The game of invention is basically distinct from today's casino slot/video games due to the fact that all possible outcomes with their corresponding, probabilities are completely known and it uses obviously manipulation-proof random number generators. However, the automatic version can be integrated into an existing video game machine where a TIMER function random number generator will be installed to replace so-called RNG software protecting casino's profit.

The derivation of some probabilities is by means of modular arithmetic, but neither the operator nor any player needs to understand it. Since all #d1(i) to #d4(i) and #d1(i,j) to #d4(i,j) are explicitly provided, no one will be required to do any modular arithmetic while the computer will just apply those numerical values. Besides, I also give examples to help everyone get acquaintance with practical calculations.

To make the bettor without regret, every hanging bet earns the credit amount equal to the payoff value of a bet up to that point. The operator can make house edge effective simply on the final payoff. Charging house edge only on final payoff makes purchasing a multi draw ticket more incentive than purchasing tickets draw by draw. Besides, house edge should be on a whole ticket instead of each single bet, and based on the ratio of payoff to the total bet amount to allow lower ratio tickets enjoy lower house edges. Naturally, setting house edges is not inventor's business, but the game's popularity depends on reasonable house edges such as by formulas provided above. The game operator shall publish how to calculate house edges. Knowing all possible outcomes with corresponding probabilities as well as knowing attractive house edges are what attract people to try to beat the house.

Besides, the operator can always by the way run contest such as follows: Anyone paying an entry fee gets a non-cashable voucher for say $1M to play. The player must make a number of certain kinds of bets, including some credit ones. Every payoff will be added to the voucher. Reaching a certain winning results will grant the player a prize, which may include some percentage of the voucher. The computer can handle contestants like regular bettors.

Due to the fact that up to the moment of a concerning draw it doesn't matter when any selection is made or changed. Thus, it can be an option that the bettor is allowed to change selections any time before the draw, or that the bettor may even purchase a ticket stating the number of certain kinds of bets without detailed selections and submit the details anytime ahead of the concerning draw. Computer random betting selections may also be made available as an option.

There are only three similar playing surfaces with ruled movements given here. But obviously the method can be applied to many other similar playing surfaces with other similar ruled movements. The number of sites and movers can easily be made different from those given above. Ruled movements can be different for different movers. Other types of betting can be added into the game such as ‘chain site’ bet, for example, the first draw moves movers #1 and #2 to site A and the next draw moves movers #3 and #4 to site B. They can also be modified by some racing characteristics such as mover #1 reaching site A ahead of mover #2 reaching site B. Chain bets can be more than four draws.

Thus, the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by examples given. 

1. A method of playing a game of chance comprising the steps of providing a plurality of moving pieces called movers on a playing surface consisting of sites, providing a plurality of ruled movements, providing a plurality of obviously manipulation-proof random number generators, such as those used at keno or lottery, one generator for each said mover, one number for each said movement while there can be more than one, but equally many copies of each number, using said generators to draw random numbers for said movers to move accordingly, permitting players to place bets on own selected said movers moving to own selected said sites by one or more rounds of drawings, providing formulae to calculate probabilities for all possible outcomes and payoff for every bet.
 2. A method of playing a game using the steps of claim 1 and further comprising the steps of providing calculation of credit for a hanging bet based on movement probability where a multiround bet is defined as hanging if it contains a selection of movers with movements matching the last round outcomes and is thus in a position to be a winner later on, permitting the hanging bet holder to use own selected percentage of the hanging bet credit to place free bets.
 3. A method of playing a game of chance using the steps of claim 1 and further by means of a video game machine or personal computer using TIMER function to randomly determine said movements, where each said movement has the probability of 1/w to be drawn if there are w distinct said movements.
 4. A method of playing a game of chance using the steps of claim 2 and further by means of a video game machine or personal computer using TIMER function to randomly determine said movements, where each said movement has the probability of 1/w to be drawn if there are w distinct said movements. 